Spectral Holography and the Computational Limit of Reality: From the Riemann Zeros to the Thermodynamic Cost of Spacetime

Here, we present a unified framework of Spectral Holography arguing that the apparent limits of physical knowledge—from quantum uncertainty to the energetic cost of computation—are symptoms of reality possessing a finite, quantifiable resolution based on a small number of degrees of freedom. We demonstrate this by connecting three disparate fields: (1) Structural Rigidity (Riemann-Pauli): The statistical repulsion of the non-trivial Riemann zeros is shown to obey the same Gaussian Unitary Ensemble (GUE) statistics that govern the repulsion of quantum energy levels (Pauli Exclusion Principle). This suggests that matter stability and abstract number theory stem from a shared matrix-based quantum "hardware" defined by approximately $N \approx 43$ degrees of freedom. (2) Epistemological Limits (Spectral Holography): The "Trumpet of Ignorance" observed in inverse quantum tomography is interpreted as a computational limit, where the available information bandwidth ($c \approx 190 \pi^2$) is finite. Uncertainty explodes not because the laws cease to exist, but because the holographic resolution "pixels out" at the boundaries. (3) Thermodynamic Cost (Landauer-Gravity): We establish that Landauer's Principle ($k_B T \ln 2$) is the fundamental thermodynamic toll required to change a single matrix bit (geometric degree of freedom). Thus, the geometry of spacetime, the logic of computation, and the structure of prime numbers are unified within a finite, information-theoretic ontology, revealing reality as a deterministic but computationally bounded matrix.